Page 25 - Textos de Matemática Vol. 34

CONNECTING BASIC BESSEL FUNCTIONS 15 (1) (2)
On the other side, the functions Jν (z; q) and Jν (z; q) satisfy a biorthogonality relation [17]
1
xJ(1)(xj(2);q2)J(2)(qxj(2);q2)d x=
ν nν ν mν q qν/2 (2) (2) 2 d(2) (2) 2
0
=−2Jν+1(jnν;q)dxJν (jnν;q)δn,m.
Another natural question is: How are these three q−Bessel functions related to (1) (2)
each other? If the functions considered are Jν (z;q) and Jν (z;q), the answer becomes very simple: Hahn [9] proved
(1.8) J(2)(z; q) = −z2/4; q J(1)(z; q) , |z| < 2. ν∞ν
(2) 2 (1)
Therefore, Jν (z; q) is the analytical continuation of −z /4; q ∞ Jν (z; q). The
only relation between Jν (z; q) and Jν (z; q) we were able to find in the literature was the Fourier-Gauss transform integral identity [4]:
∞22 J(2)(2texs;q)e2irs−s ds=qν(2−3ν)/8J(3)(q(ν−1)/4teixr;q)e−r .
νν
−∞
The purpose of the next section is to establish connection coefficient type relations
between Jν (z;q) and Jν (z;q), that is, to express functions of one type as an infinite linear combination of functions of another type.
2. Connection Formulas for the q−Bessel functions We begin by defining the auxiliary functions
(2) (3)
(3) (2)
Fν(z;q) = Gν(z;q) =
(−1)n
∞ qn2
∞ n=0
qn2 (qν+1; q; q)n
z2n, n 2 2n
(−1) (qν+1;q;q) z
and
(q; q)∞ With this notation (1.2) and (1.3) become
n=0
n
qν+1;q
∞ . J(3)(z;q) = zνA (q)G (q1 z;q)
Aν(q) =
(2.1) and
(2.2)
ννν4 ν
J(2)(z;q) = zνA (q)F (q2 z;q), ννν2